1. 15.5 Directional Derivatives
MAT 3238 Vector Calculus
15.5 Directional Derivatives

2. Homework Both written and WA HW due ???day.
I suggest you do the written part first.
Let me remind you that your only reference are
The textbook
The lecture note
Do not google or look up other references

3. Steepest Descent Algorithm
Minimize Time – path planning for a robot in Mars
Minimize Energy – choose parameters so that the energy is min.: Most stable
All kind of applications!

4. Steepest Descent Algorithm
At each time step, need to choose a direction where the changes is the greatest.
Changes is measure by slope (i.e. derivative) of the tangent along that direction.
We need a notion of slopes along different directions.

5. Derivatives Along Directions
We know how to do this in the 𝑥 and 𝑦 directions:

6. Derivatives Along Directions
How to extend this notion to other directions?

7. Directional Derivatives
The directional derivative of 𝑓 at a point in the direction of a unit vector 𝑢= 𝑎,𝑏 is if this limit exists.

8. Directional Derivatives
In particular, if... 𝑢= 1,0 , 𝐷 𝑢 𝑓 𝑥 0 , 𝑦 0 = 𝑢= 0,1 , 𝐷 𝑢 𝑓 𝑥 0 , 𝑦 0 = So, this definition is in consistent with what we know.

9. FAQ
Q: We hate limits. Can we use a formula to compute the directional derivatives?
A: Yes, of course!

10. Directional Derivatives

11. Observation #1
𝐷 𝑢 𝑓 is a linear combinations of 𝑓 𝑥 and 𝑓 𝑦 .

12. Observation #2
𝐷 𝑢 𝑓 is a linear combinations of 𝑓 𝑥 and 𝑓 𝑦 . 𝑎, 𝑏 behave “almost” like “proportions”

13. Example 1

14. Why are They Believable?
Fake Formulas?

15. The Proof

16. The Proof